On the existence of a global solution of the modified Navier–Stokes equations

We prove global existence theorems for initial-boundary value problems for the modified Navier–Stokes equations used when modeling ocean dynamic processes. First, the case of distinct vertical and horizontal viscosities for the Navier–Stokes equations is considered. Then a result due to Ladyzhenskaya for the modified Navier–Stokes equations is improved, whereby the elliptic operator is strengthened with respect to the horizontal variables alone and only for the horizontal momentum equations. Finally, the global existence and uniqueness of a solution is proved for the primitive equations describing the large-scale ocean dynamics.